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An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions

Received: 25 January 2023     Accepted: 13 February 2023     Published: 23 February 2023
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Abstract

Soliton equations are infinite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the discovery of integrability of these equations has greatly promoted the understanding of their generality, and thus promoted their rapid development. A key feature of an integrable nonlinear evolution equation is the fact that it can be expressed as the compatibility condition of two linear spectral problems, i.e., a Lax pair, which plays a crucial roles in the Darboux transformation. A major difficulty, however, is the problem of associating nonlinear evolution equations with appropriate spectral problems. Therefore, it is interesting for us to search for the new spectral problem and corresponding nonlinear evolution equations. In this paper, a new integrable nonlinear wave model and its integrable nonlinear reduction are presented by using the introduced 2 × 2 matrix spectral problem. Based on the resulting gauge transforms between the 2 × 2 matrix Lax pairs, Darboux transforms are derived for the integrable nonlinear wave model and its integrable nonlinear reduction, from which an algebraic algorithm for solving this integrable nonlinear wave model and its integrable nonlinear reduction is given. As an application of the Darboux transform, explicit exact solutions of the integrable nonlinear reduction are obtained, including solitons, breathers, and rogue waves.

Published in Applied and Computational Mathematics (Volume 12, Issue 1)
DOI 10.11648/j.acm.20231201.11
Page(s) 1-8
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

An Integrable Nonlinear Wave Model, Integrable Reduction, Darboux Transform, Exact Solutions

References
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[5] D. Levi, On a new Darboux transformation for the construction of exact solutions of the Schrödinger equation, Inverse Problems 4 (1988) 165-172.
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[9] W. X. Ma, Y. Zhou, Lump solutions to nonlinear partial differential equations via Hirota bilinear forms, J. Differential Equations 264 (2018) 2633-2659.
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[11] J. Wei, X. G. Geng, X. Zeng, The Riemann theta function solutions for the hierarchy of Bogoyavlensky lattices, Trans. Amer. Math. Soc. 371 (2019) 1483-1507.
[12] M. X. Jia, X. G. Geng, J. Wei, Algebro-geometric quasi-periodic solutions to the Bogoyavlensky lattice 2(3) equations, J. Nonlinear Sci. 32 (2022) 98.
[13] G. Mu, Z. Y. Qin, R. Grimshaw, Dynamics of rogue waves on a multisoliton background in a vector nonlinear Schrödinger equation, SIAM J. Appl. Math. 75 (2015) 1-20.
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  • APA Style

    Jingru Geng, Minna Feng. (2023). An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions. Applied and Computational Mathematics, 12(1), 1-8. https://doi.org/10.11648/j.acm.20231201.11

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    ACS Style

    Jingru Geng; Minna Feng. An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions. Appl. Comput. Math. 2023, 12(1), 1-8. doi: 10.11648/j.acm.20231201.11

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    AMA Style

    Jingru Geng, Minna Feng. An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions. Appl Comput Math. 2023;12(1):1-8. doi: 10.11648/j.acm.20231201.11

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  • @article{10.11648/j.acm.20231201.11,
      author = {Jingru Geng and Minna Feng},
      title = {An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions},
      journal = {Applied and Computational Mathematics},
      volume = {12},
      number = {1},
      pages = {1-8},
      doi = {10.11648/j.acm.20231201.11},
      url = {https://doi.org/10.11648/j.acm.20231201.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20231201.11},
      abstract = {Soliton equations are infinite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the discovery of integrability of these equations has greatly promoted the understanding of their generality, and thus promoted their rapid development. A key feature of an integrable nonlinear evolution equation is the fact that it can be expressed as the compatibility condition of two linear spectral problems, i.e., a Lax pair, which plays a crucial roles in the Darboux transformation. A major difficulty, however, is the problem of associating nonlinear evolution equations with appropriate spectral problems. Therefore, it is interesting for us to search for the new spectral problem and corresponding nonlinear evolution equations. In this paper, a new integrable nonlinear wave model and its integrable nonlinear reduction are presented by using the introduced 2 × 2 matrix spectral problem. Based on the resulting gauge transforms between the 2 × 2 matrix Lax pairs, Darboux transforms are derived for the integrable nonlinear wave model and its integrable nonlinear reduction, from which an algebraic algorithm for solving this integrable nonlinear wave model and its integrable nonlinear reduction is given. As an application of the Darboux transform, explicit exact solutions of the integrable nonlinear reduction are obtained, including solitons, breathers, and rogue waves.},
     year = {2023}
    }
    

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    T1  - An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions
    AU  - Jingru Geng
    AU  - Minna Feng
    Y1  - 2023/02/23
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    N1  - https://doi.org/10.11648/j.acm.20231201.11
    DO  - 10.11648/j.acm.20231201.11
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
    SP  - 1
    EP  - 8
    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20231201.11
    AB  - Soliton equations are infinite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the discovery of integrability of these equations has greatly promoted the understanding of their generality, and thus promoted their rapid development. A key feature of an integrable nonlinear evolution equation is the fact that it can be expressed as the compatibility condition of two linear spectral problems, i.e., a Lax pair, which plays a crucial roles in the Darboux transformation. A major difficulty, however, is the problem of associating nonlinear evolution equations with appropriate spectral problems. Therefore, it is interesting for us to search for the new spectral problem and corresponding nonlinear evolution equations. In this paper, a new integrable nonlinear wave model and its integrable nonlinear reduction are presented by using the introduced 2 × 2 matrix spectral problem. Based on the resulting gauge transforms between the 2 × 2 matrix Lax pairs, Darboux transforms are derived for the integrable nonlinear wave model and its integrable nonlinear reduction, from which an algebraic algorithm for solving this integrable nonlinear wave model and its integrable nonlinear reduction is given. As an application of the Darboux transform, explicit exact solutions of the integrable nonlinear reduction are obtained, including solitons, breathers, and rogue waves.
    VL  - 12
    IS  - 1
    ER  - 

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Author Information
  • School of Computer and Artificial Intelligence, Zhengzhou University, Zhengzhou, China

  • School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, China

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